How to Do an Indirect Proof

Indirect proofs are sort of a weird uncle of regular proofs. With an indirect proof, instead of proving that something must be true, you prove it indirectly by showing that it cannot be false.

Note the not. When your task in a proof is to prove that things are not congruent, not perpendicular, and so on, it’s a dead giveaway that you’re dealing with an indirect proof.

For the most part, an indirect proof is very similar to a regular proof. What makes it different is the way it begins and ends. And except for the beginning and end, to solve an indirect proof, you use the same techniques and theorems that you would use on regular proofs.

The best way to explain indirect proofs is by showing you an example. Here you go.

Note two peculiar things about this odd duck of a proof: the not-congruent symbols in the givens and the prove statement. The one in the prove statement is sort of what makes this an indirect proof.

Here’s a game plan showing how you can tackle this indirect proof. You assume that the prove statement is false, namely that segment PS is congruent to segment RS, and then your goal is to arrive at a contradiction of some known true thing (usually a given fact about things that are not congruent, not perpendicular, and so on). In this problem, your goal is to show that angle PQS is congruent to angle RQS, which contradicts the given.

One last thing before showing you the solution — you can write out indirect proofs in the regular format, but many geometry textbooks and teachers present indirect proofs in paragraph form, like this:

Assume the opposite of the prove statement, treating this opposite statement as a given.
Work through the problem as usual, trying to prove the opposite of one of the givens (usually the one that states something is not perpendicular, congruent, or the like).
Finish by stating that you’ve reached a contradiction and that, therefore, the prove statement must be true.

(Quod Erat Demonstrandum — “which was to be demonstrated” — for all you Latin-speakers out there; the rest of you can just say, “I’m done!”)